Inverse Design of Energy‐Absorbing Metamaterials by Topology Optimization

Abstract Compared with the forward design method through the control of geometric parameters and material types, the inverse design method based on the target stress‐strain curve is helpful for the discovery of new structures. This study proposes an optimization strategy for mechanical metamaterials based on a genetic algorithm and establishes a topology optimization method for energy‐absorbing structures with the desired stress‐strain curves. A series of structural mutation algorithms and design‐domain‐independent mesh generation method are developed to improve the efficiency of finite element analysis and optimization iteration. The algorithm realizes the design of ideal energy‐absorbing structures, which are verified by additive manufacturing and experimental characterization. The error between the stress‐strain curve of the designed structure and the target curve is less than 5%, and the densification strain reaches 0.6. Furthermore, special attention is paid to passive pedestrian protection and occupant protection, and a reasonable solution is given through the design of a multiplatform energy‐absorbing structure. The proposed topology optimization framework provides a new solution path for the elastic‐plastic large deformation problem that is unable to be resolved by using classical gradient algorithms or genetic algorithms, and simplifies the design process of energy‐absorbing mechanical metamaterials.


S1.1 Stiffness Matrix Tunable Topological Optimization Algorithm
The optimization efficiency and results of the genetic algorithm are strongly dependent on the initial population. Hence, structures with different moduli and densities are required to guide the algorithm optimization. A stiffness-matrix tunable topology optimization algorithm was proposed in our previous work, [29] where the design objective was set as the structural equivalent stiffness matrix and the constraint function was the material volume fraction.
Based on the homogenization theory, the structural equivalent stiffness matrix can be expressed as: where * refers to the equivalent stiffness matrix and φ represents the volume or area of the microstructure.
where denotes the design variable and represents the weighting factor. * and refer to the structural equivalent and target stiffness matrices, respectively. refers to the volume of grid element and represents the volume fraction of design material.
The range of the relative Young's modulus and the relative shear modulus is calculated to satisfy the Hashin-Shtrikman bounds according to the relative density of the structure during the initial design optimization. Therefore, the target stiffness matrix satisfies the Hashin-Shtrikman bounds.
Fifty structures with relative Young's modulus in the range of 0 to 0.2 and relative density in the range of 0 to 0.8 are obtained based on this algorithm, which is selected as the initial population, as shown in Figure 2aI.

S1.2 NSGAII Algorithm
The energy-absorbing structure is designed for multi-objective optimization. Hence, NSGAII in the genetic algorithm is selected as the basic algorithm. As shown in Figure S1a, it is the main flow of genetic algorithm. Elite individuals are the parent of the population and children are generated by crossover and mutation operations, as shown in Figure S1c and S1d. The children are subjected to fitness evaluation, as shown in Figure S1b, forming a non-dominated ranking. The elite individuals of the sorted children are retained as the parents of the next generation.    The discontinuous structure cannot be evaluated for fitness and exhibits limited effect on population genetic improvement. In the improved algorithm, the continuity and symmetry are calculated after the configuration crossover. The cross point is re-selected if the structure is not continuous and, if the continuous structure cannot be obtained after re-selection 50 times, the parent is re-selected to ensure that the crossover structure is effective. The random single-point mutation method adopted by the traditional algorithm makes the structure appear harmful and invalid mutations, as shown in Figure S4d, and S4e, resulting in a discontinuous or unanalyzable structure.

S1.3 Genetic Algorithms for Energy-Absorbing Topology Optimization
Therefore, an outline variation scheme for porous structures is proposed, as shown in Figure S4f. The boundary points of the structure are picked by the outline recognition algorithm, and two mutation operators, i.e., B rate1 and B rate2 , are defined. The mutation module expands the outline of structure according to the value of B rate1 and contraction the outline of the structure according to the value of B rate2 . The validity of structure is guaranteed by precise variation of the profile of porous structure.
In addition, thickness constraints are imposed to ensure that the structure is easy to manufacture. The centerline of the structure is extracted by the skeleton algorithm, as shown in Figure S5a

S2 SUPPLEMENTARY RESULTS
The design of energy absorbing structures for different target curves (curve1, curve2, curve3 and curve4) for the same platform stress was carried out to verify the diversity of the algorithm, as shown in Figure S8a. The optimization process of the optimal structure is shown in Figure S9 and the deformation process of the structure is shown in Figure S10. The design error of the structure is shown in Table S2. The given examples provide the design of ideal energy-absorbing structures and the design of functional energy-absorbing structures. The ideal energy absorption design is an energy absorption structure with different plateau stresses, i.e., σ m = 10, 20, 40, 60, 140 and 250 MPa, as shown in Figure S8b. The optimal structural optimization process for low plateau stress (σ m = 10, 20, 40 and 60 MPa) is shown in Figure 3 and the deformation process of the structure is shown in Figure S11. The optimization process of the optimal structure with high platform stress (σ m = 140 and 250 MPa) is shown in Figure S12, and the deformation process of the structure is shown in Figure S13. The simulation error of the unit cell, and simulation and experimental errors of the lattice are compared with the design target, as shown in Table S3. In the design of functional energy-absorbing structure, the optimization process for optimal structures with double platform stress is shown in Figure S14 and the deformation process of the structure is shown in Figure S15. The simulation error of the unit cell is found to be 8.38%, the simulation error of the lattice is found to be 15.34%, and the experimental error of the lattice is found to be 15.88%.
Buckling analysis was carried out to investigate the initial failure mode of the structure. The first mode of the structure is shown in Figure S16a, and the buckling eigenvalue is far greater than the initial peak force. Therefore, the structure was collapse directly rather than buckle during compression. The compression simulation results of model with imperfection and without imperfection are shown in Figure   S16b. The consistent curves of the two models also indicate that the introduction of initial imperfection factors has no effect on the structural failure process. Other structures in the manuscript were also buckling analyzed, and the same conclusion was obtained.    MPa. Figure S13. The deformation process of the optimal structure under a compressive load of a) σ m = Figure S14. The optimization process of optimal structure with double platform stress.